Question: Find $2^x$ if

\begin{align*}
2^x+3^y&=5,\\
2^{x+2}+3^{y+1} &=18.
\end{align*}
Let $2^x=a$ and $3^y=b$.  Since $2^{x+2}=2^2(2^x)$ and $3^{y+1}=3(3^y)$, the equations become

\begin{align*}
a+b&=5,\\
4a+3b&=18.
\end{align*}Multiplying the first equation by $3$ and subtracting it from the second equation, we find $a=\boxed{3}$ and $b = 2$.  Plugging these into the original equations, we find this works.